Hello. I have a definition question.
According to wikipedia and wolfram alpha, impredicativity is the self-reference of a set. What would be a simple example of this?
If we have a point on (x,f(x)), say, (a,b) and separate the two points as
so that either self-referencing result is not a function but is, at the specified value, the value itself... would this be a good example of the concept? Each result has x or f on both sides of the equals sign so that if x=a and f=b then we revert to our original values that constitute a single point.
This sounds like it is leading to the Russell paradox.
Here's an example:
I have a large library of books. Some of them have red covers. I have decided to make a catalogue of all my red books, called the Red Books Catalogue. I have decided to make this as a book, and give it a red cover. Thus, it is listed in the catalogue, as a book with a red cover.
ps. Consider the other catalogue I have made of all the books which don't contain themselves as an entry. I've made the catalogue but I cannot decide whether to list the catalogue as an entry.
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Thanks for the reply.
Do recursive statements such as the self-referencing I gave in my example have any special name, or are they just called recursive statements?
I wouldn't have called them by that name but I cannot see what is wrong with it. I don't know any better name. I would probably just have said something like: "So x is defined in a function that contains x itself." which doesn't exactly 'trip off the tongue' easily.
On-line I found these pages, which might be worth a look: